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Complex Number to Polar Form Calculator.

Convert rectangular components to modulus and principal argument.

On-device calculationNo signup
01

Set your values

Results update as you type.

Modulus: 5

Modulus

0.000000
Argument: 0.927295 rad

Argument

0.000000rad
Argument: 53.130102 °

Argument

0.000000°

Results update automatically as you type.

Use Cases

Engineering and Physics Calculations

Simplify AC circuit analysis, signal processing, and wave mechanics by converting complex impedances or phasors to polar form for magnitude and phase calculations.

Example: Convert impedance 3 + 4j to polar form to find magnitude 5 and phase 53.13°.

Mathematics Education

Verify homework or practice problems involving complex numbers, especially when learning about the polar representation and De Moivre's theorem.

Example: Check that 1 + i converts to √2 ∠ 45°.

Frequently Asked Questions

What does this calculator do?
It converts a complex number from rectangular form (a + bi) to polar form (r∠θ). You enter the real part and imaginary part, and it computes the modulus (r) and the principal argument (θ) in radians or degrees.
What is the principal argument?
The principal argument is the angle θ in the range (-π, π] (or equivalently -180° to 180°) that the complex number makes with the positive real axis. It is unique for each complex number.
Can I use this for any complex number?
Yes, as long as you provide the real and imaginary parts. The calculator handles all quadrants, including negative values, and correctly determines the angle based on the signs of the components.

Tips & Common Mistakes

Tips

  • Ensure the real and imaginary parts are entered as numbers, not expressions. Use decimal points for non-integer values.
  • Remember that the principal argument is typically given in radians or degrees; check the calculator's output format.
  • For negative real parts, the angle will be in the second or third quadrant, so the calculator adjusts the arctangent accordingly.
  • Use the polar form to easily multiply or divide complex numbers: multiply moduli and add arguments.

Common Mistakes to Avoid

  • Forgetting to include the sign of the imaginary part, which affects the quadrant and the angle.
  • Assuming the argument is always the arctangent of (imaginary/real) without considering the quadrant, leading to incorrect angles for negative real parts.
  • Mixing up degrees and radians when interpreting the result; always note the unit used by the calculator.

Last updated: August 13, 2026